Abstract

In this paper, we give strong convergence theorems for the modified S-iteration process of asymptotically quasi-nonexpansive mappings on a CAT(0) space which extend and improve many results in the literature.

MSC: Primary 47H09; secondary 47H10.

Keywords

1 Introduction

A metric space X is a CAT(0)space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. The initials of the term ‘CAT’ are in honor of E. Cartan, A. D. Alexanderov and V. A. Toponogov. A CAT(0) space is a generalization of the Hadamard manifold, which is a simply connected, complete Riemannian manifold such that the sectional curvature is non-positive. In fact, it is very well known that any complete simply connected Riemannian manifold with non-positive sectional curvature is a CAT(0) space. The complex Hilbert ball with a hyperbolic metric is a CAT(0) space (see [1]). Other examples include Pre-Hilbert spaces, R-trees (see [2]) and Euclidean buildings (see [3]). A CAT(0) space plays a fundamental role in various areas of mathematics (see Bridson and Haefliger [2], Burago, Burago and Ivanov [4], Gromov [5]). Moreover, there are applications in biology and computer science as well [6, 7].

Fixed point theory in a CAT(0) space has been first studied by Kirk (see [8, 9]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory in a CAT(0) space has been rapidly developed and many papers have appeared (see, e.g., [8–14]). It is worth mentioning that the results in a CAT(0) space can be applied to any CAT(k) space with k≤0 since any CAT(k) space is a CAT(k′) space for every k′≥k (see [[2], p.165]). Throughout the paper, ℕ and ℝ denote the set of natural numbers and the set of real numbers, respectively.

The Mann iteration process is defined by the sequence {xn},

{x1∈K,xn+1=(1−an)xn+anTxn,n∈N,

(1.1)

where {an} is a sequence in (0,1).

Further, the Ishikawa iteration process is defined as the sequence {xn},

{x1∈K,xn+1=(1−an)xn+anTyn,yn=(1−bn)xn+bnTxn,n∈N,

(1.2)

where {an} and {bn} are the sequences in (0,1). This iteration process reduces to the Mann iteration process when bn=0 for all n∈N.

Agarwal, O’Regan and Sahu [15] introduced the S-iteration process in a Banach space,

{x1∈K,xn+1=(1−an)Txn+anTyn,yn=(1−bn)xn+bnTxn,n∈N,

(1.3)

where {an} and {bn} are the sequences in (0,1). Note that (1.3) is independent of (1.2) (and hence of (1.1)). They showed that their process is independent of those of Mann and Ishikawa and converges faster than both of these (see [[15], Proposition 3.1]).

Schu [16], in 1991, considered the modified Mann iteration process which is a generalization of the Mann iteration process,

{x1∈K,xn+1=(1−an)xn+anTnxn,n∈N,

(1.4)

where {an} is a sequence in (0,1).

Tan and Xu [17], in 1994, studied the modified Ishikawa iteration process which is a generalization of the Ishikawa iteration process,

{x1∈K,xn+1=(1−an)xn+anTnyn,yn=(1−bn)xn+bnTnxn,n∈N,

(1.5)

where the sequences {an} and {bn} are in (0,1). This iteration process reduces to the modified Mann iteration process when bn=0 for all n∈N.

where the sequences {an} and {bn} are in (0,1). Note that (1.6) is independent of (1.5) (and hence of (1.4)). Also, (1.6) reduces to (1.3) when n=1.

We now modify (1.6) in a CAT(0) space as follows.

Let K be a nonempty closed convex subset of a complete CAT(0) space X and T:K→K be an asymptotically quasi-nonexpansive mapping with F(T)≠∅. Suppose that {xn} is a sequence generated iteratively by

{x1∈K,xn+1=(1−an)Tnxn⊕anTnyn,yn=(1−bn)xn⊕bnTnxn,n∈N,

(1.7)

where and throughout the paper {an}, {bn} are the sequences such that 0≤an,bn≤1 for all n∈N.

In this paper, we study the modified S-iteration process for asymptotically quasi-nonexpansive mappings on the CAT(0) space and generalize some results of Khan and Abbas [14] which studied the S-iteration process in a CAT(0) space for nonexpansive mappings. This paper contains three sections. In Section 2, we first collect some known preliminaries and lemmas that will be used in the proofs of our main theorems. We give the main results related to the strong convergence theorems of the modified S-iteration process in a CAT(0) space in Section 3. Under some suitable condition, we obtain the main theorems which state that {xn} converges strongly to a fixed point of T. Our results can be applied to an S-iteration process since the modified S-iteration process reduces to the S-iteration process when n=1.

2 Preliminaries and lemmas

Let us recall some definitions and known results in the existing literature on this concept.

Let (X,d) be a metric space and K its nonempty subset. Let T:K→K be a mapping. A point x∈K is called a fixed point of T if Tx=x. We will also denote by F(T) the set of fixed points of T, that is, F(T)={x∈K:Tx=x}.

The concept of quasi-nonexpansiveness was introduced by Diaz and Metcalf [18] in 1967, the concept of asymptotically nonexpansiveness was introduced by Goebel and Kirk [19] in 1972. The iterative approximation problems for asymptotically quasi-nonexpansive mapping were studied by Liu [20], Fukhar-ud-din et al. [21], Khan et al. [22] and Beg et al. [23] in a Banach space and a CAT(0) space.

Definition 1 Let (X,d) be a metric space and K be its nonempty subset. Then T:K→K is said to be

(1)

nonexpansive if d(Tx,Ty)≤d(x,y) for all x,y∈K,

(2)

asymptotically nonexpansive if there exists a sequence {un}∈[0,∞) with the property limn→∞un=0 and such that d(Tnx,Tny)≤(1+un)d(x,y) for all x,y∈K,

(3)

quasi-nonexpansive if d(Tx,p)≤d(x,p) for all x∈K, p∈F(T),

(4)

asymptotically quasi-nonexpansive if there exists a sequence {un}∈[0,∞) with the property limn→∞un=0 and such that d(Tnx,p)≤(1+un)d(x,p) for all x∈K, p∈F(T),

(5)

semi-compact if for a sequence {xn} in K with limn→∞d(xn,Txn)=0, there exists a subsequence {xnk} of {xn} such that xnk→p∈K.

Remark 1 From Definition 1, it is clear that the class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings includes nonexpansive mappings, whereas the class of asymptotically quasi-nonexpansive mappings is larger than that of quasi-nonexpansive mappings and asymptotically nonexpansive mappings. The reverse of these implications may not be true.

Let (X,d) be a metric space. A geodesic path joining x∈X to y∈X (or more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]⊂R to X such that c(0)=x, c(l)=y and d(c(t),c(t′))=|t−t′| for all t,t′∈[0,l]. In particular, c is an isometry and d(x,y)=l. The image of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [x,y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x to y for each x,y∈X. A subset Y⊂X is said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic triangle△(x1,x2,x3) in a geodesic metric space (X,d) consists of three points in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparisontriangle for a geodesic triangle △(x1,x2,x3) in (X,d) is a triangle △¯(x1,x2,x3)=△(x¯1,x¯2,x¯3) in the Euclidean plane R2 such that

Finally, we observe that if x, y1, y2 are points of a CAT(0) space and if y0 is the midpoint of the segment [y1,y2], which we will denote by y1⊕y22, then the CAT(0) inequality implies

d(x,y1⊕y22)2≤12d(x,y1)2+12d(x,y2)2−14d(y1,y2)2.

(2.1)

The equality holds for the Euclidean metric. In fact (see [[2], p.163]), a geodesic metric space is a CAT(0) space if and only if it satisfies inequality (2.1) (which is known as the CN inequality of Bruhat and Tits [25]).

Let x,y∈X, by [[12], Lemma 2.1(iv)] for each t∈[0,1], then there exists a unique point z∈[x,y] such that

d(x,z)=td(x,y),d(y,z)=(1−t)d(x,y).

(2.2)

From now on, we will use the notation (1−t)x⊕ty for the unique point z satisfying (2.2). By using this notation, Dhompongsa and Panyanak [12] obtained the following lemma which will be used frequently in the proof of our main results.

for all m,n>n1. Therefore, {xn} is a Cauchy sequence in K. Since the set K is complete, the sequence {xn} must be convergence to a point in K. Let limn→∞xn=p∈K. Here after, we show that p is a fixed point. By limn→∞xn=p, for all ϵ1>0, there exists n2∈N such that

d(xn,p)<ϵ12(2+u1)

(3.4)

for all n>n2. From (3.3), for each ϵ1>0, there exists n3∈N such that

d(xn,F(T))<ϵ12(4+3u1)

for all n>n3. In particular, inf{d(xn3,p):p∈F(T)}<ϵ12(4+3u1). Thus, there must exist p⋆∈F(T) such that

Since ϵ1 is arbitrary, so d(Tp,p)=0, i.e., Tp=p. Therefore, p∈F(T). This completes the proof. □

Remark 2 Let the hypothesis of Theorem 1 be satisfied and T:K→K be an asymptotically nonexpansive or quasi-nonexpansive mapping. From Remark 1, the class of asymptotically quasi-nonexpansive mappings includes quasi-nonexpansive mappings and asymptotically nonexpansive mappings. Then the sequence {xn} converges strongly to a fixed point of T.

Now, we give the following corollaries which have been proved by Theorem 1.

Corollary 1Under the hypothesis of Theorem 1, Tsatisfies the following conditions:

(1)

limn→∞d(xn,Txn)=0.

(2)

If the sequence{zn}inKsatisfieslimn→∞d(zn,Tzn)=0, then

lim infn→∞d(zn,F(T))=0orlim supn→∞d(zn,F(T))=0.

Then the sequence{xn}converges strongly to a fixed point ofT.

Proof It follows from the hypothesis that limn→∞d(xn,Txn)=0. From (2),

lim infn→∞d(xn,F(T))=0orlim supn→∞d(xn,F(T))=0.

Therefore, the sequence {xn} must converge to a fixed point of T by Theorem 1. □

Corollary 2Under the hypothesis of Theorem 1, Tsatisfies the following conditions:

(1)

limn→∞d(xn,Txn)=0.

(2)

There exists a functionf:[0,∞)→[0,∞)which is right-continuous at 0, f(0)=0andf(r)>0for allr>0such that

d(x,Tx)≥f(d(x,F(T)))for allx∈K,

whered(x,F(T))=infz∈F(T)d(x,z).

Then the sequence{xn}converges strongly to a fixed point ofT.

Proof It follows from the hypothesis that

limn→∞f(d(xn,F(T))≤limn→∞d(xn,Txn)=0.

That is,

limn→∞f(d(xn,F(T))=0.

Since f:[0,∞)→[0,∞) is right-continuous at 0 and f(0)=0, therefore we have

limn→∞d(xn,F(T))=0.

Thus, lim infn→∞d(xn,F(T))=lim supn→∞d(xn,F(T))=0. By Theorem 1, the sequence {xn} converges strongly to q, a fixed point of T. This completes the proof. □

Finally, we give the following theorem which has a different hypothesis from Theorem 1.

Proof From the hypothesis, we have limn→∞d(xn,Txn)=0. Also, since T is semi-compact, there exists a subsequence {xnk} of {xn} such that xnk→p∈K. Hence,

d(p,Tp)=limn→∞d(xnk,Txnk)=0.

Thus, p∈F(T). By (3.2),

d(xn+1,p)≤(1+un)2d(xn,p)=(1+2un+un2)d(xn,p).

Since ∑n=1∞un<∞, we have ∑n=1∞(2un+un2)<∞. By Lemma 2, limn→∞d(xn,p) exists and xnk→p∈F(T) gives that xn→p∈F(T). This completes the proof. □

4 Conclusions

The class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings includes nonexpansive mappings, where as the class of asymptotically quasi-nonexpansive mappings is larger than that of quasi-nonexpansive mappings and asymptotically nonexpansive mappings. Then these results presented in this paper extend and generalize some works for a CAT(0) space in the literature.

Declarations

Acknowledgements

This paper has been presented in The ‘International Conference on Applied Analysis and Algebra (ICAAA2012)’ in Yıldız Technical University, 20-24 June 2012, Istanbul, Turkey.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, Faculty of Sciences and Arts, Sakarya University

Copyright

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